If |x + 1/2| = |y + 1/2|, what is the value of x + y?

Question

If  |x+\frac{1}{2}|=|y+\frac{1}{2}| , what is the value of  x+y ?

(1)  xy<0

(2)  x>0  and  y<0

stne · Answer

|x+\frac{1}{2}|=|y+\frac{1}{2}|

Both sides can be  +ve

Both sides can be  -ve

RHS can be  +ve   and LHS can be  -ve

RHS can be  -ve  and LHS can be  +ve

All the above cases can be summarised in just two cases :

(i) Consider both sides  +ve  ( this is equivalent to considering both sides negative )
(ii) Consider both sides to be of different signs.

If both sides are  +ve  then we get:

x+\frac{1}{2}= y+\frac{1}{2}

x=y

If both sides are of different signs then we get:

x+\frac{1}{2}= -y-\frac{1}{2}

x+y=-1

(1) xy<0

This case is not possible if  x=y 
Hence it must be that both the sides are of different signs and  x+y =-1

SUFF.

(2) x>0 and y<0

Similar to statement (1) This case is not possible if  x=y. 
Hence it must be that both the sides are of different signs and  x+y =-1

SUFF.

Ans D

Hope it helps.

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